A method for solving LSM problems of small size in the AHP

The Analytic Hierarchy Process (AHP) is one of the most popular methods used in Multi-Attribute Decision Making. It provides ratio-scale measurements of the priorities of elements on the various levels of a hierarchy. These priorities are obtained through the pairwise comparisons of elements on one level with reference to each element on the immediate higher level. The Eigenvector Method (EM) and some distance minimizing methods such as the Least Squares Method (LSM), Logarithmic Least Squares Method (LLSM), Weighted Least Squares Method (WLSM) and Chi Squared Method (χ²M) are the tools for computing the priorities of the alternatives. This paper studies a method for generating all the solutions of the LSM problem for 3 × 3 matrices. We observe non-uniqueness and rank reversals by presenting numerical results.